Question

How do you find the domain and range of \[h(x) = \ln (x - 6)\]?

Answer 1

Hint: Given equation \[h(x) = \ln (x - 6)\]
We know that for the logarithmic function to exist the function under the log must be greater than 0 and also base value should be greater than 0. And we can find the range from the graph of the function. Firstly we will define the range of log then its whole equation range. In this way, we can get the full domain very easily.

Complete step by step solution:
We know to find the domain of the logarithmic function, we have to find the function greater than 0 which is inside the log.
For the given question,
\[\begin{array}{l}
x - 6 > 0\\
 \Rightarrow x > 6
\end{array}\]
So, the domain of the function \[h(x) = \ln (x - 6)\] will be \[(6, + \infty ){\rm{ }}or\;\{ x \vdots x > 6\} \].
To find range:
 We know the range of logarithmic function that is \[\log x{\rm{ }}\] is from \[{\rm{ }}( - \infty , + \infty )\]. The graph of the given function that is \[h(x) = \ln (x - 6)\] is just like the graph of \[\log x\]
just it will be shifted by 6 units towards +x-axis. Instead of passing through (1,0) it will pass through (7,0).

So we can conclude that the range of \[h(x) = \ln (x - 6)\] will be from \[( - \infty , + \infty )\].

Note: The function inside the log must be greater than 0 and also base value should be greater than 0 and must not be equal to 1. And for finding range we can also use the hit and trial method. . For conventional type methods, we can first define the domain of the inside number, and then we can apply any type of function to achieve its domain just like in the case of sin x where the sin function limits the domain as well as the range of x.